International Journal of Engineering & Technology IJET-IJENS Vol:12 No:05
83
Minimization of THD and Angle Calculation for
Multilevel Inverters
Carlos Alberto Lozano Espinosa, Member, IEEE, Ivonne Portocarrero, and Mauricio Izquierdo

Abstract—
This paper shows an algorithm to calculate the
switching angles of a cascaded multilevel inverter minimizing the
total harmonic distortion. The implementation uses a cascaded
multilevel inverter with only one battery feeding one bridge and
one transformer for each switching angle and connected in
cascade with the other transformers. A comparison of total
harmonic distortion (THD) with selective harmonic elimination
technique and the angles calculation with this algorithm is
shown.
Index Term— Multilevel inverter, switching angle calculation,
THD minimization, selective harmonic elimination.
I. INTRODUCTION
The objective of a power inverter is to have as little harmonic
content as possible. Several techniques has been demonstrated
to reduce in a certain way this content, by eliminating some
harmonics to place a low- pass filter with a further cut
frequency or to reduce the total harmonic distortion, as a good
indicator of the inverter performance [1].
Other papers concentrate their efforts in minimizing the THD
equation by approximating to some results in order to have a
fast calculation of the switching angles [2], [6], [7], [8], or the
calculation of the angles based on the elimination of some
selective harmonics [3], [4].
This paper starts demonstrating that after calculation of the
angle for eliminating one harmonics in a three level inverter
(SHE type inverter with one harmonics elimination) the THD
could be minimize by a simple formula. Minimization of THD
in SHE inverters with more switching angles can use the same
THD equation. An equation for THD calculation in multilevel
inverters with more than three levels, is presented. Based on
this equation an algorithm to find the minimum value of THD
is proposed. The implementation of the cascaded multilevel
inverter with optimum values for switching angles and
minimum THD uses a single DC battery with a transformer at
the output of each bridge [5]. Five bridges are implemented to
form an eleven steps inverter with all steps of equal value, so
that, as shown later, only one battery will be used to generate
each step. Some works shows techniques to use only one
battery in cascaded inverters [9].
II. THD CALCULATION FOR SHE AND MULTILEVEL
INVERTERS
A. Harmonic Elimination of SHE inverters
Selective Harmonic Elimination (SHE) technique for one
phase inverters is one of the options for inverters to reduce
some harmonics and set the cut frequency of a low- pass filter
with a higher value to reduce the size of inductances and
capacitances of the filter. Even some harmonics can be
eliminated, the total harmonic distortion could increase.
Fig. 1. SHE inverter with elimination of one harmonic
In a SHE inverter with one harmonic elimination, shown in
the Fourier coefficients are,
( )
(1)
Fig. 1,
To eliminate the third harmonic,
and THD is,
)
( )
√ (
( )
Fig. 2 shows THD vs. switching angle of the SHE inverter
with one harmonic elimination. The minimum THD value
occurs in
with THD of 28.96%.
Carlos Alberto Lozano Espinosa is with the department of Electronics and
Computing Science of the Pontificia Universidad Javeriana, Santiago de Cali,
Valle del Cauca, Colombia (e-mail: carlosal@javerianacali.edu.co)
Ivonne Portocarrero is with the department of Electronics and Computing
Science of the Pontificia Universidad Javeriana, Santiago de Cali, Valle del
Cauca, Colombia (ivonne@javerianacali.edu.co)
Mauricio Izquierdo is with the department of Electronics and Computing
Science of the Pontificia Universidad Javeriana, Santiago de Cali, Valle del
Cauca, Colombia (e-mail: mizquierdo@javerianacali.edu.co)
1211905-8282-IJET-IJENS © October 2012 IJENS
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International Journal of Engineering & Technology IJET-IJENS Vol:12 No:05
84
Fig. 2. THD vs. switching angle for single harmonic
elimination in a SHE inverter
The Fourier coefficients of the SHE inverter for two
harmonic elimination are,
(
)
(
))
(
(2)
To eliminate the third and the fifth harmonics, the following
equations must be satisfied:
(
)
(
)
(
)
(
)
(3)
Solving the above equations,
With these angles THD is,
√ (
(
) (
(
)
and
)
(
(
.
))
(4)
)
THD does not improve with more angles added to eliminate
more harmonics in SHE inverters, thus the lowest THD that
can be obtain is 28.96% with one angle of 23.2°.
B. Harmonic Elimination in Multilevel Inverter
Multilevel inverters provide a less THD than other inverters
and it can improve with more levels added. One of the
drawbacks is the calculation of the switching angles since the
more levels are needed, more angles must be calculated and
more time is spent in calculation. Fig. 3 shows a multilevel
inverter output for eleven steps. The RMS voltage for (2p+1)
levels is,
√
∑
(
)
(5)
Where p is the number of switching angles in half cycle.
The Fourier coefficients are,
∑
( )
∫
∑
(
)
(6)
One of the most used techniques for finding the switching
angles is to use the Fourier coefficients to eliminate some
harmonics. The number of harmonics to be eliminated is equal
to the number of switching angles to be calculated minus one,
with this technique.
Fig. 3. Multilevel inverter with eleven steps
For example, for five switching angles to eliminate the fifth,
seventh, eleventh and thirteenth harmonics, the resultant
equations to calculate switching angles are,
(
(
(
(
(
)
)
)
(
)
)
(
(
)
)
(
(
)
( )
)
)
(
(
)
)
(
(
(
(
(
)
)
)
(
(
)
)
(
(
(
)
)
)
)
)
(
(
)
)
(7)
The first equation of (7) has the number four at the right of
the equation to express the amplitude of the coefficient of the
fundamental component in the Fourier series. Switching
angles can be obtained by iterating with Newton- Raphson
method. The obtained switching angles are
,
,
,
, and
. THD
for p switching angles is,
√
∑
(
∑
)
(∑
(
))
( )
(8)
For this particular example with switching angles like
above, THD= 7.93%.
III. THD MINIMIZATION IN MULTILEVEL INVERTER
Equation (8) can be used to minimize THD assuming
, for
. A computer program
can be used to find the switching angles for the minimum
THD using the same equation, nevertheless the amount of
time of calculation increases with the number of angles.
Calculation of THD requires computing of p cosines, two
square roots, 2p summations, p+2 multiplications, and one
division, where p is the number of angles to be calculated.
Calculation of the minimum THD depends on angle
resolution. For one degree resolution the first angle goes from
1° to 89° in steps of 1°, the second angle goes from 2° to 89°
)
and so on, so that for p angles it is needed ∏ (
THD calculations. For example, for two angles with one
degree resolution it is needed 3,916 THD calculations,
assuming the program checks for the minimum THD in all
possible values of each angle. For three angles with the same
resolution it is needed 113,564 THD calculations. The
flowchart of the program is shown in Fig. 4. The more
switching angles are needed, the more for loops must be
nested and the program can spent a lot of time running. That is
1211905-8282-IJET-IJENS © October 2012 IJENS
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International Journal of Engineering & Technology IJET-IJENS Vol:12 No:05
why the increment (Inc) is a key to find first a close value for
switching angles and then modify the limit values of each
loop, that is, angle i that goes from aiL to aiH, its limit values.
The increment (Inc) starts with a large value, say 6 in the all
range of the angles, from 0 to 90 degrees. The first run of the
program gives the seeds or close angle values for the
minimum THD. For example, for p= 5 and Inc= 6 the first run
of the program gives 7°, 14°, 27°, 40° and 59°. This first run
needs 2002 THD calculations, less than is needed for two
angles calculation. The new limit values can be calculated
from the Inc value as aiL= aix-Inc/2 and aiH= aix+Inc/2,
where aix is the new value of the angle i. For this example the
limit values are (4, 10), (11, 17), (24, 30), (37, 43) and (56,
62). Closer angles for the real minimum THD can be found
with Inc equal to 1. In this case the second run of the program
gives 6°, 17°, 29°, 42° and 60°, and the new limit values for
Inc equal to 0.1 are (5.5, 6.5), (16.5, 17.5), (28.5, 29.5), (41.5,
42.5) and (59.5, 60.5). The second run of the program needs
7,776 THD calculations. The third run of the program with
these values gives the angles in table I for eleven levels.
85
IV. IMPLEMENTATION
For testing the switching angle calculation a cascaded
multilevel inverter of eleven steps was used. This inverter has
only one battery with five boards with the same
characteristics, one 1- phase full wave inverter bridge with a
transformer at the output. This schematic diagram is shown in
Fig. 5. Each bridge has its own switching angle. For example,
for bridge one the output V1 is of a single inverter with angle
. The output of each bridge is added in cascade to form the
multilevel shape of eleven steps.
Fig. 5. Cascaded multilevel inverter.
Each transformer used in the inverter is of 1:2.7 for a 12
volt battery, giving 110 volts RMS, 60 hertz. Fig. 6 shows an
oscilloscope view of the inverter output.
Fig. 4. Flowchart for switching angles calculation
A list of switching angles for up to fifteen levels is
presented in table 1.
TABLE I
SWITCHING ANGLES FOR MINIMUM THD IN MULTILEVEL INVERTERS
Levels
THD
(%)
3
5
7
9
11
13
15
28.96
16.42
11.53
8.90
7.26
6.13
5.31
23.2°
12.8°
8.9°
6.8°
5.5°
4.6°
4.0°
41.8°
27.6°
20.8°
16.7°
13.9°
12.0°
Fig. 6. Multilevel inverter voltage output .
50.5°
36.2°
28.6°
23.7°
20.3°
55.8°
42.1°
34.2°
29.0°
59.5°
46.3°
38.6°
62.1°
49.7°
V. CONCLUSION
64.3°
With THD minimization using this program, for five angles,
THD is 7.26%, compared with five harmonic elimination
method above that has THD in 7.93%.
A flowchart of a program for THD minimization is shown
to calculate the switching angles of a cascaded multilevel
inverter based on the equation of THD for multilevel inverters.
Compared with harmonic elimination, the THD resulted from
switching angle calculation with this algorithm is lower. A list
of switching angles for up to fifteen levels is presented from
1211905-8282-IJET-IJENS © October 2012 IJENS
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International Journal of Engineering & Technology IJET-IJENS Vol:12 No:05
86
the algorithm described. This algorithm could be used for online calculation of the minimum THD [6] in applications
where some conditions change or systems with feedback
control. As compared with other THD minimization
techniques [7], [8], this algorithm guarantees the minimum
THD possible with a fast calculation. The implementation in a
cascaded multilevel inverter shows a simple way to get the
desired output voltage with minimum THD using a single
battery.
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